#323133

order	512
nilpotency class	3
isoclinism	isoclinic class of #323133
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2541
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3205	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

676 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2,4,8,5,9,13,7)(3,6,12,15,11,14,10,16)(17,18)(19,25)(20,32)(21,24)(22,27)(23,28)(26,33)(29,37)(30,44)(31,39)(34,42)(35,38)(36,41)(40,43)(45,47)(46,48);(1,3,4,12,5,11,13,10)(2,6,8,15,9,14,7,16)(18,24,28,27)(19,29,31,36)(25,39)(26,38,42,32)(30,43,45,46)(37,41)(40,48)(44,47);(2,7)(3,10)(4,13)(6,14)(8,9)(11,12)(17,19)(18,25)(20,30)(21,29)(22,36)(23,31)(24,37)(26,40)(27,41)(28,39)(32,44)(33,43)(34,46)(35,45)(38,47)(42,48);(3,11)(6,14)(10,12)(15,16)(17,20,21,33,23,35,22,34)(18,26,27,32,28,42,24,38)(19,30,29,43,31,45,36,46)(25,40,41,44,39,48,37,47)

Polycyclic presentation:

2^9|a^2=f,b^2=eg,d^2=e,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=di,f^c=fi,g^c=ghi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 230329);